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Jun-Muk Hwang

Publications and source records attributed to Jun-Muk Hwang.

At least 55 records · Page 3Linked to original sources
arXiv2010

Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity

The prolongation g^{(k)} of a linear Lie algebra g \subset gl(V) plays an important role in the study of symmetries of G-structures. Cartan and Kobayashi-Nagano have given a complete classification of irreducible linear Lie algebras g \subset gl(V) with non-zero prolongations. If g is the Lie algebra aut(\hat{S}) of infinitesimal linear automorphisms of a projective variety S \subset \BP V, its prolongation g^{(k)} is related to the symmetries of cone structures, an important example of which is the variety of minimal rational tangents in the study of uniruled projective manifolds. From this perspective, understanding the prolongation aut(\hat{S})^{(k)} is useful in questions related to the automorphism groups of uniruled projective manifolds. Our main result is a complete classification of irreducible non-degenerate nonsingular variety with non zero prolongations, which can be viewed as a generalization of the result of Cartan and Kobayashi-Nagano. As an application, we show that when $S$ is linearly normal and Sec(S) \neq P(V), the blow-up of P(V) along S has the target rigidity property, i.e., any deformation of a surjective morphism Y \to Bl_S(PV) comes from the automorphisms of Bl_S(PV).

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arXiv2010

Local structure of principally polarized stable Lagrangian fibrations

A holomorphic Lagrangian fibration is stable if the characteristic cycles of the singular fibers are of type $I_m, 1 \leq m <\infty,$ or $A_{\infty}$. We will give a complete description of the local structure of a stable Lagrangian fibration when it is principally polarized. In particular, we give an explicit form of the period map of such a fibration and conversely, for a period map of the described type, we construct a principally polarized stable Lagrangian fibration with the given period map. This enables us to give a number of examples exhibiting interesting behavior of the characteristic cycles.

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arXiv2010

Buser-Sarnak invariant and projective normality of abelian varieties

We show that a general $n$-dimensional polarized abelian variety $(A,L)$ of a given polarization type and satisfying $ h^0(A, L) \geq \dfrac{8^n}{2} \cdot \dfrac{n^n}{n !}$ is projectively normal. In the process, we also obtain a sharp lower bound for the volume of a purely one-dimensional complex analytic subvariety in a geodesic tubular neighborhood of a subtorus of a compact complex torus.

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arXiv2009

Minimal rational curves on complete toric manifolds

We shall show that the variety of minimal rational tangents on a complete toric manifold X is linear and minimal components in RatCurves^n(X) corresponds bijectively to some special primitive collections in the fan defining X.

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arXiv2009

Unobstructedness of deformations of holomorphic maps onto Fano manifolds of Picard number 1

We show that deformations of a surjective morphism onto a Fano manifold of Picard number 1 are unobstructed and rigid modulo the automorphisms of the target, if the variety of minimal rational tangents of the Fano manifold is non-linear or finite. The condition on the variety of minimal rational tangents holds for practically all known examples of Fano manifolds of Picard number 1, except the projective space. When the variety of minimal rational tangents is non-linear, the proof is based on an earlier result of N. Mok and the author on the birationality of the tangent map. When the varieties of minimal rational tangents of the Fano manifold is finite, the key idea is to factorize the given surjective morphism, after some transformation, through a universal morphism associated to the minimal rational curves.

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arXiv2009

Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents

We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety $Z$, a family of minimal rational curves with $Z$-isotrivial varieties of minimal rational tangents is locally equivalent to the flat model. We show that this is the case when $Z$ satisfies certain projective-geometric conditions, which hold for a non-singular hypersurface of degree $\geq 4$.

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arXiv2009

Multiple fibers of holomorphic Lagrangian fibrations

We determine all possible multiplicities of general singular fibers of a holomorphic Lagrangian fibration, under the assumption that all components of the fibers are of Fujiki class. The multiplicities are at most 6 and the possible values are intricately related to the Kodaira type of the characteristic cycle.

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arXiv2008

Quasi-complete homogeneous contact manifold associated to a cubic form

Starting from a cubic form, we give a general construction of a quasi-complete homogeneous manifold endowed with a natural contact structure. We show that it can be compactified into a projective contact manifold if and only if the cubic form is the determinant of a simple cubic Jordan algebra.

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arXiv2008

Characteristic foliation on a hypersurface of general type in a projective symplectic manifold

Given a projective symplectic manifold $M$ and a non-singular hypersurface $X \subset M$, the symplectic form of $M$ induces a foliation of rank 1 on $X$, called the characteristic foliation. We study the question when the characteristic foliation is algebraic, namely, all the leaves are algebraic curves. Our main result is that the characteristic foliation of $X$ is not algebraic if $X$ is of general type. For the proof, we first establish an étale version of Reeb stability theorem in foliation theory and then combine it with the positivity of the direct image sheaves associated to families of curves.

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arXiv2008

Compact Manifolds Covered by a Torus

Let $X$ be a connected compact complex manifold admitting a finite surjective map $A \to X$ from a complex torus $A.$ We prove that up to finite étale cover, $X$ is a product of projective spaces and a torus.

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arXiv2007

Base manifolds for fibrations of projective irreducible symplectic manifolds

Given a projective irreducible symplectic manifold $M$ of dimension $2n$, a projective manifold $X$ and a surjective holomorphic map $f:M \to X$ with connected fibers of positive dimension, we prove that $X$ is biholomorphic to the projective space of dimension $n$. The proof is obtained by exploiting two geometric structures at general points of $X$: the affine structure arising from the action variables of the Lagrangian fibration $f$ and the structure defined by the variety of minimal rational tangents on the Fano manifold $X$.

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arXiv2007

Characteristic foliation on the discriminantal hypersurface of a holomorphic Lagrangian fibration

We give a Kodaira-type classification of general singular fibers of a holomorphic Lagrangian fibration in Fujiki's class $\mathcal C$. Our approach is based on the study of the characteristic vector field of the discriminantal hypersurface, which naturally arises from the defining equation of the hypersurface via the symplectic form. As an application, we show that the characteristic foliation of the discriminantal hypersurface has algebraic leaves which are either rational curves or smooth elliptic curves.

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arXiv2005

Holomorphic maps onto varieties of non-negative Kodaira dimension

A classical result in complex geometry says that the automorphism group of a manifold of general type is discrete. It is more generally true that there are only finitely many surjective morphisms between two fixed projective manifolds of general type. Rigidity of surjective morphisms, and the failure of a morphism to be rigid have been studied by a number of authors in the past. The main result of this paper states that surjective morphisms are always rigid, unless there is a clear geometric reason for it. More precisely, we can say the following. First, deformations of surjective morphisms between normal projective varieties are unobstructed unless the target variety is covered by rational curves. Second, if the target is not covered by rational curves, then surjective morphisms are infinitesimally rigid, unless the morphism factors via a variety with positive-dimensional automorphism group. In this case, the Hom-scheme can be completely described.

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arXiv2004

On the volumes of complex hyperbolic manifolds with cusps

We study the problem of bounding the number of cusps of a complex hyperbolic manifold in terms of its volume. Applying algebro-geometric methods using Mumford's work on toroidal compactifications and its generalization due to N. Mok and W.-K. To, we get a bound which is considerably better than those obtained previously by methods of geometric topology.

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arXiv2004

A bound on the number of curves of a given degree through a general point of a projective variety

Let $X$ be an irreducible projective variety of dimension $n$ in a projective space and let $x$ be a point of $X$. Denote by ${\rm Curves}_d(X,x)$ the space of curves of degree $d$ lying on $X$ and passing through $x$. We will show that the number of components of ${\rm Curves}_d(X,x)$ for any smooth point $x$ outside a subvariety of codimension $\geq 2$ is bounded by a number depending only on $n$ and $d$. An effective bound is given. A key ingredient of the proof is an argument from Ein-Küchle-Lazarsfeld's work on Seshadri numbers.

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arXiv2004

Geometry of chains of minimal rational curves

Chains of minimal degree rational curves have been used as an important tool in the study of Fano manifolds. Their own geometric properties, however, have not been studied much. The goal of the paper is to introduce an infinitesimal method to study chains of minimal rational curves via varieties of minimal rational tangents and their higher secants. For many examples of Fano manifolds this method can be used to compute the minimal length of chains needed to join two general points. One consequence of our computation is a bound on the multiplicities of divisors at a general point of the moduli of stable bundles of rank two on a curve.

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